Stable three-dimensional vortex solitons of high topological charge in a Rydberg-dressed Bose-Einstein condensate with spin-orbit coupling

Stable vortex solitons (VSs) are objects of great interest for fundamental studies and various applications, including particle trapping, microscopy, data encoding, and matter-wave gyroscopes. However, three-dimensional (3D) VSs with high topological charges, supported by self-attractive nonlinearities, are unstable against fragmentation, which eventually leads to internal blowup (supercritical collapse) of the fragments. Here, we propose a scheme for realizing stable 3D VSs with topological charges up to 5 and 6 in the two components of a binary, Rydberg-dressed Bose-Einstein condensate with spin-orbit coupling (SOC). We show that, if the SOC strength exceeds a critical value, the rotational symmetry of the VSs in the transverse plane gets broken, resulting in a separation of the two components. Nevertheless, the VSs with the broken symmetry remain stable. The VS stability domains are identified in the system's parameter space. Moreover, application of torque to the stable VSs sets them in the state of robust gyroscopic precession.

Comparing the winding numbers of two one-dimensional two-band topological systems by their wavefunction overlap

The measurement of topological numbers is crucial in the research of topological systems. In this article, we propose a protocol for obtaining the topological number (specifically, winding numbers in this case) of an unknown one-dimensional (1D) two-band topological system by comparing it with a known topological system. We consider two 1D two-band topological systems and their Bloch wavefunction overlap and verify a theorem. This theorem states that when the momentum varies by 2π, the number of cycles during which the magnitude of the wavefunction overlap varies from 0 to 1 and then back to 0 is equal to the absolute value of the difference between the topological numbers of these two systems. Furthermore, we propose two experimental schemes, one in a cold atom system and another one in a qubit system, which offer convenient and robust measurement methods for determining topological numbers of unknown states through quenching.

Non-Abelian lattice gauge fields in photonic synthetic frequency dimensions

Non-Abelian gauge fields1 provide a conceptual framework to describe particles having spins, underlying many phenomena in electrodynamics, condensed-matter physics2,3 and particle physics4,5. Lattice models6 of non-Abelian gauge fields allow us to understand their physical implications in extended systems. The theoretical importance of non-Abelian lattice gauge fields motivates their experimental synthesis and explorations7-9. Photons are fundamental particles for which artificial gauge fields can be synthesized10-30, yet the demonstration of non-Abelian lattice gauge fields for photons has not been achieved. Here we demonstrate SU(2) lattice gauge fields for photons in the synthetic frequency dimensions31,32, a playground to study lattice physics in a scalable and programmable way. In our lattice model, we theoretically observe that homogeneous non-Abelian lattice gauge potentials induce Dirac cones at time-reversal-invariant momenta in the Brillouin zone. We experimentally confirm the presence of non-Abelian lattice gauge fields by two signatures: linear band crossings at the Dirac cones, and the associated direction reversal of eigenstate trajectories. We further demonstrate a non-Abelian scalar lattice gauge potential that lifts the degeneracies of the Dirac cones. Our results highlight the implications of non-Abelian lattice gauge fields in topological physics, and provide a starting point for demonstrations of emerging non-Abelian physics in the photonic synthetic dimensions. Our results may also benefit photonic technologies by providing controls of photon spins and pseudo-spins in topologically non-trivial ways33.

Counterdiabatic Driving for Periodically Driven Systems

Periodically driven systems have emerged as a useful technique to engineer the properties of quantum systems, and are in the process of being developed into a standard toolbox for quantum simulation. An outstanding challenge that leaves this toolbox incomplete is the manipulation of the states dressed by strong periodic drives. The state-of-the-art in Floquet control is the adiabatic change of parameters. Yet, this requires long protocols conflicting with the limited coherence times in experiments. To achieve fast control of nonequilibrium quantum matter, we generalize the notion of variational counterdiabatic driving away from equilibrium focusing on Floquet systems. We derive a nonperturbative variational principle to find local approximations to the adiabatic gauge potential for the effective Floquet Hamiltonian. It enables transitionless driving of Floquet eigenstates far away from the adiabatic regime. We discuss applications to two-level, Floquet band, and interacting periodically driven models. The developed technique allows us to capture nonperturbative photon resonances and obtain high-fidelity protocols that respect experimental limitations like the locality of the accessible control terms.

Semivortex solitons and their excited states in spin-orbit-coupled binary bosonic condensates

It is known that two-dimensional two-component fundamental solitons of the semivortex (SV) type, with vorticities (s_{+},s_{-})=(0,1) in their components, are stable ground states (GSs) in the spin-orbit-coupled (SOC) binary Bose-Einstein condensate with the contact self-attraction acting in both components, in spite of the possibility of the critical collapse in the system. However, excited states (ESs) of the SV solitons, with the vorticity set (s_{+},s_{-})=(S_{+},S_{+}+1) and S_{+}=1,2,3,..., are unstable in the same system. We construct ESs of SV solitons in the SOC system with opposite signs of the self-interaction in the two components. The main finding is stability of the ES-SV solitons, with the extra vorticity (at least) up to S_{+}=6. The threshold value of the norm for the onset of the critical collapse, N_{thr}, in these excited states is higher than the commonly known critical value, N_{c}≈5.85, associated with the single-component Townes solitons, N_{thr} increasing with the growth of S_{+}. A velocity interval for stable motion of the GS-SV solitons is found too. The results suggest a solution for the challenging problem of the creation of stable vortex solitons with high topological charges.

Dissipation-Induced Extended-Localized Transition

A mobility edge (ME), representing the critical energy that distinguishes between extended and localized states, is a key concept in understanding the transition between extended (metallic) and localized (insulating) states in disordered and quasiperiodic systems. Here we explore the impact of dissipation on a quasiperiodic system featuring MEs by calculating steady-state density matrix and analyzing quench dynamics with sudden introduction of dissipation. We demonstrate that dissipation can lead the system into specific states predominantly characterized by either extended or localized states, irrespective of the initial state. Our results establish the use of dissipation as a new avenue for inducing transitions between extended and localized states and for manipulating dynamic behaviors of particles.

Topological spin-orbit-coupled fermions beyond rotating wave approximation

The realization of spin-orbit-coupled ultracold gases has driven a wide range of research and is typically based on the rotating wave approximation (RWA). By neglecting the counter-rotating terms, RWA characterizes a single near-resonant spin-orbit (SO) coupling in a two-level system. Here, we propose and experimentally realize a new scheme for achieving a pair of two-dimensional (2D) SO couplings for ultracold fermions beyond RWA. This work not only realizes the first anomalous Floquet topological Fermi gas beyond RWA, but also significantly improves the lifetime of the 2D-SO-coupled Fermi gas. Based on pump-probe quench measurements, we observe a deterministic phase relation between two sets of SO couplings, which is characteristic of our beyond-RWA scheme and enables the two SO couplings to be simultaneously tuned to the optimum 2D configurations. We observe intriguing band topology by measuring two-ring band-inversion surfaces, quantitatively consistent with a Floquet topological Fermi gas in the regime of high Chern numbers. Our study can open an avenue to explore exotic SO physics and anomalous topological states based on long-lived SO-coupled ultracold fermions.

Bright solitons in a spin-orbit-coupled dipolar Bose-Einstein condensate trapped within a double-lattice

By effectively controlling the dipole-dipole interaction, we investigate the characteristics of the ground state of bright solitons in a spin-orbit coupled dipolar Bose-Einstein condensate. The dipolar atoms are trapped within a double-lattice which consists of a linear and a nonlinear lattice. We derive the motion equations of the different spin components, taking the controlling mechanisms of the dipole-dipole interaction into account. An analytical expression of dipole-dipole interaction is derived. By adjusting the dipole polarization angle, the dipole interaction can be adjusted from attraction to repulsion. On this basis, we study the generation and manipulation of the bright solitons using both the analytical variational method and numerical imaginary time evolution. The stability of the bright solitons is also analyzed and we map out the stability phase diagram. By adjusting the long-range dipole-dipole interaction, one can achieve manipulation of bright solitons in all aspects, including the existence, width, nodes, and stability. Considering the complexity of our system, our results will have enormous potential applications in quantum simulation of complex systems.

Three-dimensional solitons in Rydberg-dressed cold atomic gases with spin-orbit coupling

We present numerical results for three-dimensional (3D) solitons with symmetries of the semi-vortex (SV) and mixed-mode (MM) types, which can be created in spinor Bose-Einstein condensates of Rydberg atoms under the action of the spin-orbit coupling (SOC). By means of systematic numerical computations, we demonstrate that the interplay of SOC and long-range spherically symmetric Rydberg interactions stabilize the 3D solitons, improving their resistance to collapse. We find how the stability range depends on the strengths of the SOC and Rydberg interactions and the soft-core atomic radius.

Berry Curvature and Bulk-Boundary Correspondence from Transport Measurement for Photonic Chern Bands

Berry curvature is a fundamental element to characterize topological quantum physics, while a full measurement of Berry curvature in momentum space was not reported for topological states. Here we achieve two-dimensional Berry curvature reconstruction in a photonic quantum anomalous Hall system via Hall transport measurement of a momentum-resolved wave packet. Integrating measured Berry curvature over the two-dimensional Brillouin zone, we obtain Chern numbers corresponding to -1 and 0. Further, we identify bulk-boundary correspondence by measuring topology-linked chiral edge states at the boundary. The full topological characterization of photonic Chern bands from Berry curvature, Chern number, and edge transport measurements enables our photonic system to serve as a versatile platform for further in-depth study of novel topological physics.

Ultrafast Electronic Dynamics in Anisotropic Indirect Interlayer Excitonic States of Monolayer WSe2/ReS2 Heterojunctions

Understanding ultrafast electronic dynamics of the interlayer excitonic states in atomically thin transition metal dichalcogenides is of importance in engineering valleytronics and developing excitonic integrated circuits. In this work, we experimentally explored the ultrafast dynamics of indirect interlayer excitonic states in monolayer type II WSe2/ReS2 heterojunctions using time-resolved photoemission electron microscopy, which reveals its anisotropic behavior. The ultrafast cooling and decay of excited-state electrons exhibit significant linear dichroism. The ab initio theoretical calculations provide unambiguous evidence that this linear dichroism result is primarily associated with the anisotropic nonradiative recombination of indirect interlayer excitonic states. Measuring time-resolved photoemission energy spectra, we have further revealed the ultrafast evolution of excited-state electrons in anisotropic indirect interlayer excitonic states. The findings have important implications for controlling the interlayer moiré excitonic effects and designing anisotropic optoelectronic devices.

Matter-wave solitons in an array of spin-orbit-coupled Bose-Einstein condensates

We investigate matter-wave solitons in a binary Bose-Einstein condensate (BEC) with spin-orbit (SO) coupling, loaded in a one-dimensional (1D) deep optical lattice and a three-dimensional anisotropic magnetic trap, which creates an array of elongated sub-BECs with transverse tunneling. We show that the system supports 1D continuous and discrete solitons localized in the longitudinal (along the array) and the transverse (across the array) directions, respectively. In addition, such solitons are always unpolarized in the zero-momentum state but polarized in finite-momentum states. We also show that the system supports stable two-dimensional semidiscrete solitons, including single- and multiple-peaked ones, localized in both the longitudinal and transverse directions. Stability diagrams for single-peaked semidiscrete solitons in different parameter spaces are identified. The results reported here are beneficial not only for understanding the physical property of SO-coupled BECs but also for generating new types of matter-wave solitons.

Non-Abelian Inverse Anderson Transitions

Inverse Anderson transitions, where the flat-band localization is destroyed by disorder, have been wildly investigated in quantum and classical systems in the presence of Abelian gauge fields. Here, we report the first investigation on inverse Anderson transitions in the system with non-Abelian gauge fields. It is found that pseudospin-dependent localized and delocalized eigenstates coexist in the disordered non-Abelian Aharonov-Bohm cage, making inverse Anderson transitions depend on the relative phase of two internal pseudospins. Such an exotic phenomenon induced by the interplay between non-Abelian gauge fields and disorder has no Abelian analogy. Furthermore, we theoretically design and experimentally fabricate non-Abelian Aharonov-Bohm topolectrical circuits to observe the non-Abelian inverse Anderson transition. Through the direct measurements of frequency-dependent impedance responses and voltage dynamics, the pseudospin-dependent non-Abelian inverse Anderson transitions are observed. Our results establish the connection between inverse Anderson transitions and non-Abelian gauge fields, and thus comprise a new insight on the fundamental aspects of localization in disordered non-Abelian flat-band systems.

Optomechanics and quantum phase of the Bose-Einstein condensate with the cavity mediated spin-orbit coupling

We investigated the optomechanical dynamics and explored the quantum phase of a Bose-Einstein condensate in a ring cavity. The interaction between the atoms and the cavity field in the running wave mode induces a semiquantized spin-orbit coupling (SOC) for the atoms. We found that the evolution of the magnetic excitations of the matter field resembles that of an optomechanical oscillator moving in a viscous optical medium, with very good integrability and traceability, regardless of the atomic interaction. Moreover, the light-atom coupling induces a sign-changeable long-range interatomic interaction, which reshapes the typical energy spectrum of the system in a drastic manner. As a result, a new quantum phase featuring a high quantum degeneracy was found in the transitional area for SOC. Our scheme is immediately realizable and the results are measurable in experiments.

Raman laser induced self-organization with topology in a dipolar condensate

We investigate the ground states of a dipolar Bose-Einstein condensate (BEC) subject to Raman laser induced spin-orbit coupling with mean-field theory. Owing to the interplay between spin-orbit coupling and atom-atom interactions, the BEC presents remarkable self-organization behavior and thus hosts various exotic phases including vortex with discrete rotational symmetry, stripe with spin helix, and chiral lattices with C₄ symmetry. The peculiar chiral self-organized array of square lattice, which spontaneously breaks both U(1) and rotational symmetries, is observed when the contact interaction is considerable in comparison with the spin-orbit coupling. Moreover, we show that the Raman-induced spin-orbit coupling plays a crucial role in forming rich topological spin textures of the chiral self-organized phases by introducing a channel for atoms to turn on spin flipping between two components. The self-organization phenomena predicted here feature topology owing to spin-orbit coupling. In addition, we find long-lived metastable self-organized arrays with C₆ symmetry in the case of strong spin-orbit coupling. We also present a proposal to observe these predicted phases in ultracold atomic dipolar gases with laser-induced spin-orbit coupling, which may stimulate broad theoretical as well as experimental interest.

Tuning Anomalous Floquet Topological Bands with Ultracold Atoms

The Floquet engineering opens the way to create new topological states without counterparts in static systems. Here, we report the experimental realization and characterization of new anomalous topological states with high-precision Floquet engineering for ultracold atoms trapped in a shaking optical Raman lattice. The Floquet band topology is manipulated by tuning the driving-induced band crossings referred to as band inversion surfaces (BISs), whose configurations fully characterize the topology of the underlying states. We uncover various exotic anomalous topological states by measuring the configurations of BISs that correspond to the bulk Floquet topology. In particular, we identify an unprecedented anomalous Floquet valley-Hall state that possesses anomalous helical-like edge modes protected by valleys and a chiral state with high Chern number.

Dipolar physics: a review of experiments with magnetic quantum gases

Since the achievement of quantum degeneracy in gases of chromium atoms in 2004, the experimental investigation of ultracold gases made of highly magnetic atoms has blossomed. The field has yielded the observation of many unprecedented phenomena, in particular those in which long-range and anisotropic dipole-dipole interactions (DDIs) play a crucial role. In this review, we aim to present the aspects of the magnetic quantum-gas platform that make it unique for exploring ultracold and quantum physics as well as to give a thorough overview of experimental achievements. Highly magnetic atoms distinguish themselves by the fact that their electronic ground-state configuration possesses a large electronic total angular momentum. This results in a large magnetic moment and a rich electronic transition spectrum. Such transitions are useful for cooling, trapping, and manipulating these atoms. The complex atomic structure and large dipolar moments of these atoms also lead to a dense spectrum of resonances in their two-body scattering behaviour. These resonances can be used to control the interatomic interactions and, in particular, the relative importance of contact over dipolar interactions. These features provide exquisite control knobs for exploring the few- and many-body physics of dipolar quantum gases. The study of dipolar effects in magnetic quantum gases has covered various few-body phenomena that are based on elastic and inelastic anisotropic scattering. Various many-body effects have also been demonstrated. These affect both the shape, stability, dynamics, and excitations of fully polarised repulsive Bose or Fermi gases. Beyond the mean-field instability, strong dipolar interactions competing with slightly weaker contact interactions between magnetic bosons yield new quantum-stabilised states, among which are self-bound droplets, droplet assemblies, and supersolids. Dipolar interactions also deeply affect the physics of atomic gases with an internal degree of freedom as these interactions intrinsically couple spin and atomic motion. Finally, long-range dipolar interactions can stabilise strongly correlated excited states of 1D gases and also impact the physics of lattice-confined systems, both at the spin-polarised level (Hubbard models with off-site interactions) and at the spinful level (XYZ models). In the present manuscript, we aim to provide an extensive overview of the various related experimental achievements up to the present.

Effective potentials in a rotating spin-orbit-coupled spin-1 spinor condensate

We theoretically study the stationary-state vortex lattice configurations of rotating spin-orbit (SO)- and coherently-coupled spin-1 Bose-Einstein condensates (BECs) trapped in quasi-two-dimensional harmonic potentials. The combined effects of rotation, SO and coherent couplings are analyzed systematically from the single-particle perspective. Through the single-particle Hamiltonian, which is exactly solvable for one-dimensional coupling, we illustrate that a boson in these rotating SO- and coherently-coupled condensates are subjected to effective toroidal, symmetric double-well, or asymmetric double-well potentials under specific coupling and rotation strengths. In the presence of mean-field interactions, using the coupled Gross-Pitaevskii formalism at moderate to high rotation frequencies, the analytically obtained effective potential minima and the numerically obtained coarse-grained density maxima position are in excellent agreement. On rapid rotation, we further find that the spin-expectation per particle of an antiferromagnetic spin-1 BEC approaches unity indicating a similarity in the response with ferromagnetic SO-coupled condensates.

Synthetic Topological Vacua of Yang-Mills Fields in Bose-Einstein Condensates

Topological vacua are a family of degenerate ground states of Yang-Mills fields with zero field strength but nontrivial topological structures. They play a fundamental role in particle physics and quantum field theory, but have not yet been experimentally observed. Here we report the first theoretical proposal and experimental realization of synthetic topological vacua with a cloud of atomic Bose-Einstein condensates. Our setup provides a promising platform to demonstrate the fundamental concept that a vacuum, rather than being empty, has rich spatial structures. The Hamiltonian for the vacuum of topological number n=1 is synthesized and the related Hopf index is measured. The vacuum of topological number n=2 is also realized, and we find that vacua with different topological numbers have distinctive spin textures and Hopf links. Our Letter opens up opportunities for exploring topological vacua and related long-sought-after instantons in tabletop experiments.

Observing a topological phase transition with deep neural networks from experimental images of ultracold atoms

Although classifying topological quantum phases have attracted great interests, the absence of local order parameter generically makes it challenging to detect a topological phase transition from experimental data. Recent advances in machine learning algorithms enable physicists to analyze experimental data with unprecedented high sensitivities, and identify quantum phases even in the presence of unavoidable noises. Here, we report a successful identification of topological phase transitions using a deep convolutional neural network trained with low signal-to-noise-ratio (SNR) experimental data obtained in a symmetry-protected topological system of spin-orbit-coupled fermions. We apply the trained network to unseen data to map out a whole phase diagram, which predicts the positions of the two topological phase transitions that are consistent with the results obtained by using the conventional method on higher SNR data. By visualizing the filters and post-convolutional results of the convolutional layer, we further find that the CNN uses the same information to make the classification in the system as the conventional analysis, namely spin imbalance, but with an advantage concerning SNR. Our work highlights the potential of machine learning techniques to be used in various quantum systems.

Wave Packet Dynamics in Synthetic Non-Abelian Gauge Fields

It is generally admitted that in quantum mechanics, the electromagnetic potentials have physical interpretations otherwise absent in classical physics as illustrated by the Aharonov-Bohm effect. In 1984, Berry interpreted this effect as a geometrical phase factor. The same year, Wilczek and Zee generalized the concept of Berry phases to degenerate levels and showed that a non-Abelian gauge field arises in these systems. In sharp contrast with the Abelian case, spatially uniform non-Abelian gauge fields can induce particle noninertial motion. We explore this intriguing phenomenon with a degenerated Fermionic atomic gas subject to a two-dimensional synthetic SU(2) non-Abelian gauge field. We reveal the spin Hall nature of the noninertial dynamic as well as its anisotropy in amplitude and frequency due to the spin texture of the system. We finally draw the similarities and differences of the observed wave packet dynamic and the celebrated Zitterbewegung effect of the relativistic Dirac equation.

Free-Fermionic Topological Quantum Sensors

Second order quantum phase transitions, with well-known features such as long-range entanglement, symmetry breaking, and gap closing, exhibit quantum enhancement for sensing at criticality. However, it is unclear which of these features are responsible for this enhancement. To address this issue, we investigate phase transitions in free-fermionic topological systems that exhibit neither symmetry-breaking nor long-range entanglement. We analytically demonstrate that quantum enhanced sensing is possible using topological edge states near the phase boundary. Remarkably, such enhancement also endures for ground states of such models that are accessible in solid state experiments. We illustrate the results with 1D Su-Schrieffer-Heeger chain and a 2D Chern insulator which are both experimentally accessible. While neither symmetry-breaking nor long-range entanglement are essential, gap closing remains as the major candidate for the ultimate source of quantum enhanced sensing. In addition, we also provide a fixed and simple measurement strategy that achieves near-optimal precision for sensing using generic edge states irrespective of the parameter value. This paves the way for development of topological quantum sensors which are expected to also be robust against local perturbations.

Floquet Engineering a Bosonic Josephson Junction

We study Floquet engineering of the tunnel coupling between a pair of one-dimensional bosonic quasicondensates in a tilted double-well potential. By modulating the energy difference between the two wells, we reestablish tunnel coupling and precisely control its amplitude and phase. This allows us to initiate coherence between two initially uncorrelated Bose gases and prepare different initial states in the emerging sine-Gordon Hamiltonian. We fully characterize the Floquet system and study the dependence of both equilibrium properties and relaxation on the modulation.

Domain-Wall Topology Induced by Spontaneous Symmetry Breaking in Polariton Graphene

We present a numerical study of exciton-polariton (polariton) condensation in a staggered polariton graphene showing a gapped s band. The condensation occurs at the kinetically favorable negative mass extrema (K and K^{'} valleys) of the valence band. Considering attractive polariton-polariton interaction allows us to generate a spatially extended condensate. The symmetry breaking occurring during the condensate buildup leads to the formation of valley-polarized domains. This process can either be spontaneous, following the Kibble-Zurek scenario, or triggered, leading to a controlled spatial distribution of valley-polarized domains. The selection of a single valley breaks time-reversal symmetry, and the walls separating domains exhibit a reconfigurable topologically protected chiral current. This current emerges as a result of the interplay between the nontrivial valley topology and the condensation-induced symmetry breaking.

Nonlinear localized modes in one-dimensional nanoscale dark-state optical lattices

Optical lattices (OLs) with conventional spatial periodic λ/2, formed by interfering the counterpropagating laser beams with wavelength λ, are versatile tools to study the dynamical and static properties of ultracold atoms. OLs with subwavelength spatial structure have been realized in recent quantum-gas experiment, offering new possibility for nonlinear and quantum control of ultracold atoms at the nano scale. Herein, we study theoretically and numerically the formation, property, and dynamics of matter-wave localized gap modes of Bose-Einstein condensates loaded in a one-dimensional nanoscale dark-state OL consisted of an array of optical subwavelength barriers. The nonlinear localized modes, in the forms of on- and off-site fundamental gap solitons, and dipole ones, are demonstrated; and we uncover that, counterintuitively, these modes exhibit always a cusplike (side peaks) mode even for a deeply subwavelength adiabatic lattice, contrary to the previously reported results in conventional deep OLs where the localized gap modes are highly confined in a single lattice cell. The (in)stability features of all the predicted localized modes are verified through the linear-stability analysis and direct perturbed simulations. Our predicted results are attainable in current ultracold atoms experiments with the cutting-edge technique, pushing the nonlinear control of ultracold atoms with short-period OLs as an enabling technology into subwavelength structures.

Topological Spin Texture of Chiral Edge States in Photonic Two-Dimensional Quantum Walks

Topological insulators host topology-linked boundary states, whose spin and charge degrees of freedom could be exploited to design topological devices with enhanced functionality. We experimentally observe that dissipationless chiral edge states in a spin-orbit coupled anomalous Floquet topological phase exhibit topological spin texture on boundaries, realized via a two-dimensional quantum walk. Our experiment shows that, for a walker traveling around a closed loop along the boundary in real space, its spin evolves and winds through a great circle on the Bloch sphere, which implies that edge-spin texture has nontrivial winding. This topological spin winding is protected by a chiral-like symmetry emerging for the low-energy Hamiltonian. Our experiment confirms that two-dimensional anomalous Floquet topological systems exhibit topological spin texture on the boundary, which could inspire novel topology-based spintronic phenomena and devices.

Majorana corner states in an attractive quantum spin Hall insulator with opposite in-plane Zeeman energy at two sublattice sites

Higher-order topological superconductors and superfluids (SFs) host lower-dimensional Majorana corner and hinge states since novel topology exhibitions on boundaries. While such topological nontrivial phases have been explored extensively, more possible schemes are necessary for engineering Majorana states. In this paper we propose Majorana corner states could be realized in a two-dimensional attractive quantum spin-Hall insulator with opposite in-plane Zeeman energy at two sublattice sites. The appropriate Zeeman field leads to the opposite Dirac mass for adjacent edges of a square sample, and naturally induce Majorana corner states. This topological phase can be characterized by Majorana edge polarizations, and it is robust against perturbations on random potentials and random phase fluctuations as long as the edge gap remains open. Our work provides a new possibility to realize a second-order topological SF in two dimensions and engineer Majorana corner states.

Second-order topological insulator in periodically driven optical lattices

The higher-order topological insulator (HOTI) is a new type of topological system which has special bulk-edge correspondence compared with conventional topological insulators. In this work, we propose a scheme to realize Floquet HOTI with ultracold atoms in optical lattices. With the combination of periodically spin-dependent driving of the superlattices and a long-range coupling term, a Floquet second-order topological insulator with four zero-energy corner states emerges, whose Wannier bands are gapless and exhibit interesting bulk topology. Furthermore, the nearest-neighbor anisotropic coupling term also induced other intriguing topological phenomena, e.g. non-topologically protected corner states and topological semimetal for two different types of lattice structures respectively. Our scheme may give insight into the construction of different types of higher-order topological insulators in synthetic systems. It also provides an experimentally feasible platform to research the relations between different types of topological states and may have a wide range of applications in future.

Experimental Realization of a Fermionic Spin-Momentum Lattice

We experimentally realize a spin-momentum lattice with a homogeneously trapped Fermi gas. The lattice is created via cyclically rotated atom-laser couplings between three bare atomic spin states, and are such that they form a triangular lattice in a synthetic spin-momentum space. We demonstrate the lattice and explore its dynamics with spin- and momentum-resolved absorption imaging. This platform will provide new opportunities for synthetic spin systems and the engineering of topological bands. In particular, the use of three spin states in two spatial dimensions would allow the simulation of synthetic magnetic fields of high spatial uniformity, which would lead to ultranarrow Chern bands that support robust fractional quantum Hall states.

Non-Hermitian Spatial Symmetries and Their Stabilized Normal and Exceptional Topological Semimetals

We study non-Hermitian spatial symmetries-a class of symmetries that have no counterparts in Hermitian systems-and study how normal and exceptional semimetals can be stabilized by these symmetries. Different from internal ones, spatial symmetries act nonlocally in momentum space and enforce global constraints on both band degeneracies and topological quantities at different locations. In deriving general constraints on band degeneracies and topological invariants, we demonstrate that non-Hermitian spatial symmetries are on an equal footing with, but are essentially different from Hermitian ones. First, we discover the nonlocal Hermitian conjugate pair of exceptional or normal band degeneracies that are enforced by non-Hermitian spatial symmetries. Remarkably, we find that these pairs lead to the symmetry-enforced violation of the Fermion doubling theorem in the long-time limit. Second, with the topological constraints, we unravel that a certain exceptional manifold is only compatible with and stabilized by non-Hermitian spatial symmetries but is intrinsically incompatible with Hermitian spatial symmetries. We illustrate these findings using two three-dimensional models of a non-Hermitian Weyl semimetal and an exceptional unconventional Weyl semimetal. Experimental cold-atom realizations of both models are also proposed.

Topological insulators on the square-hexagon lattice driven by next-nearest-neighbor hopping

We investigate the topological phase transition of the square-hexagon lattice driven by the next-nearest-neighbor (NNN) hopping. By means of the Fukui-Hatsugai method, the topological invariantZ₂can be determined. The phase diagrams in the (t₁,t₂) plane for different filling fractions are displayed, together with the size of the bulk band gap. We find the competition betweent₁andt₂can drive the system into topological nontrivial phase, withZ₂= 1. Interestingly, for 2/5 and 3/5 filling fractions, topological nontrivial phase can be easily realized when the NNN hoppings are turned on. Besides, the phase diagrams in the plane oft₂andλ_so2(t₁andλ_so1) are also investigated. By numerically diagonalizing the Hamiltonian, the bulk band structures are calculated. And the topological trivial and nontrivial phase are also distinguished in terms of helical edge state. In experiments, these topological phase transitions may be realized by shaking optical lattice.

"Designing synthetic topological matter with atoms and lights"

One of the most interesting directions in quantum simulations with ultracold atoms is the expansion of our capability to investigate exotic topological matter. Using sophisticated atom-light couplings in an atomic system, scientists have demonstrated several iconic lattice models that exhibit non-trivial band topology in a controlled manner.

Spin-orbit coupling in buckled monolayer nitrogene

Buckled monolayer nitrogene has been recently predicted to be stable above the room temperature. The low atomic number of nitrogen atom suggests, that spin–orbit coupling in nitrogene is weak, similar to graphene or silicene. We employ first principles calculations and perform a systematic study of the intrinsic and extrinsic spin–orbit coupling in this material. We calculate the spin mixing parameter \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b^2$$\end{document}b2, reflecting the strength of the intrinsic spin–orbit coupling and find, that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b^2$$\end{document}b2 is relatively small, on the order of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$10^{-6}$$\end{document}10-6. It also displays a weak anisotropy, opposite for electrons and holes. To study extrinsic effects of spin–orbit coupling we apply a transverse electric field enabling spin–orbit fields \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega$$\end{document}Ω. We find, that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega$$\end{document}Ω are on the order of a single \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu$$\end{document}μeV in the valence band, and tens to a hundred of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu$$\end{document}μeV in the conduction band, depending on the applied electric field. Similar to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b^2$$\end{document}b2, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega$$\end{document}Ω is also anisotropic, in particular for the conduction electrons.

Atom-optically synthetic gauge fields for a noninteracting Bose gas

Synthetic gauge fields in synthetic dimensions are now of great interest. This concept provides a convenient manner for exploring topological phases of matter. Here, we report on the first experimental realization of an atom-optically synthetic gauge field based on the synthetic momentum-state lattice of a Bose gas of ¹³³Cs atoms, where magnetically controlled Feshbach resonance is used to tune the interacting lattice into noninteracting regime. Specifically, we engineer a noninteracting one-dimensional lattice into a two-leg ladder with tunable synthetic gauge fields. We observe the flux-dependent populations of atoms and measure the gauge field-induced chiral currents in the two legs. We also show that an inhomogeneous gauge field could control the atomic transport in the ladder. Our results lay the groundwork for using a clean noninteracting synthetic momentum-state lattice to study the gauge field-induced topological physics.

Localization and spin dynamics of spin-orbit-coupled Bose-Einstein condensates in deep optical lattices

We analytically and numerically discuss the dynamics of two pseudospin components Bose-Einstein condensates (BECs) with spin-orbit coupling (SOC) in deep optical lattices. Rich localized phenomena, such as breathers, solitons, self-trapping, and diffusion, are revealed and strongly depend on the strength of the atomic interaction, SOC, Raman detuning, and the spin polarization (i.e., the initial population difference of atoms between the two pseudospin components of BECs). The critical conditions for the transition of localized states are derived analytically. Based on the critical conditions, the detailed dynamical phase diagram describing the different dynamical regimes is derived. When the Raman detuning satisfies a critical condition, localized states with a fixed initial spin polarization can be observed. When the critical condition is not satisfied, we use two quenching methods, i.e., suddenly and linearly quenching Raman detuning from the soliton or breather state, to discuss the spin dynamics, phase transition, and wave packet dynamics by numerical simulation. The sudden quenching results in a damped oscillation of spin polarization and transforms the system to a new polarized state. Interestingly, the linear quenching of Raman detuning induces a controllable phase transition from an unpolarized phase to an expected polarized phase, while the soliton or breather dynamics is maintained.

Exotic Vortex States with Discrete Rotational Symmetry in Atomic Fermi Gases with Spin-Orbital-Angular-Momentum Coupling

We investigate the superfluidity of a two-component Fermi gas with spin-orbital-angular-momentum coupling (SOAMC). Because of the intricate interplay of SOAMC, two-photon detuning and atom-atom interaction, a family of vortex ground states emerges in a broad parameter regime of the phase diagram, in contrast to the usual case where an external rotation or magnetic field is generally required. More strikingly, an unprecedented vortex state, which breaks the continuous rotational symmetry to a discrete one spontaneously, is predicted to occur. The underlying physics are elucidated and verified by numerical simulations. The unique density distributions of the predicted vortex states enable a direct observation in experiment.

Localization on a Synthetic Hall Cylinder

By engineering laser-atom interactions, both Hall ribbons and Hall cylinders as fundamental theoretical tools in condensed matter physics have recently been synthesized in laboratories. Here, we show that turning a synthetic Hall ribbon into a synthetic Hall cylinder could naturally lead to localization. Unlike a Hall ribbon, a Hall cylinder hosts an intrinsic lattice, which arises due to the periodic boundary condition in the azimuthal direction, in addition to the external periodic potential imposed by extra lasers. When these two lattices are incommensurate, localization may occur on a synthetic Hall cylinder. Near the localization-delocalization transitions, physical observables strongly depend on the axial magnetic flux, providing us a sensitive means to probe either the transition or the axial flux using one another. In the irrational limit, physical observables are no longer affected by the axial flux, signifying a scheme to suppress decoherence induced by fluctuations of the axial flux.

Realization of an ideal Weyl semimetal band in a quantum gas with 3D spin-orbit coupling

Weyl semimetals are three-dimensional (3D) gapless topological phases with Weyl cones in the bulk band. According to lattice theory, Weyl cones must come in pairs, with the minimum number of cones being two. A semimetal with only two Weyl cones is an ideal Weyl semimetal (IWSM). Here we report the experimental realization of an IWSM band by engineering 3D spin-orbit coupling for ultracold atoms. The topological Weyl points are clearly measured via the virtual slicing imaging technique in equilibrium and are further resolved in the quench dynamics. The realization of an IWSM band opens an avenue to investigate various exotic phenomena that are difficult to access in solids.

Ground-state phase and superfluidity of tunable spin-orbit-coupled Bose-Einstein condensates

We theoretically study the ground-state phases and superfluidity of tunable spin-orbit-coupled Bose-Einstein condensates (BECs) under the periodic driving of Raman coupling. An effective time-independent Floquet Hamiltonian is proposed by using a high-frequency approximation, and we find single-particle dispersion, spin-orbit-coupling, and asymmetrical nonlinear two-body interaction can be modulated effectively by the periodic driving. The critical Raman coupling characterizing the phase transition and relevant physical quantities in three different phases (the stripe phase, plane-wave phase, and zero momentum phase) are obtained analytically. Our results indicate that the boundary of ground-state phases can be controlled and the system will undergo three different phase transitions by adjusting the external driving. Interestingly, we find the contrast of the stripe density can be enhanced by the periodic driving in the stripe phase. We also study the superfluidity of tunable spin-orbit-coupled BECs and find the dynamical instability can be tuned by the periodic driving of Raman coupling. Furthermore, the sound velocity of the ground-state and superfluidity state can be controlled effectively by tuning the periodic driving strength. Our results indicate that the periodic driving of Raman coupling provides a powerful tool to manipulate the ground-state phase transition and dynamical instability of spin-orbit-coupled BECs.

Topological features without a lattice in Rashba spin-orbit coupled atoms

Topological order can be found in a wide range of physical systems, from crystalline solids, photonic meta-materials and even atmospheric waves to optomechanic, acoustic and atomic systems. Topological systems are a robust foundation for creating quantized channels for transporting electrical current, light, and atmospheric disturbances. These topological effects are quantified in terms of integer-valued 'invariants', such as the Chern number, applicable to the quantum Hall effect, or the [Formula: see text] invariant suitable for topological insulators. Here, we report the engineering of Rashba spin-orbit coupling for a cold atomic gas giving non-trivial topology, without the underlying crystalline structure that conventionally yields integer Chern numbers. We validated our procedure by spectroscopically measuring both branches of the Rashba dispersion relation which touch at a single Dirac point. We then measured the quantum geometry underlying the dispersion relation using matter-wave interferometry to implement a form of quantum state tomography, giving a Berry's phase with magnitude π. This implies that opening a gap at the Dirac point would give two dispersions (bands) each with half-integer Chern number, potentially implying new forms of topological transport.

Emergence of the Unconventional Type-II Nambu-Goldstone Modes with Topological Origin in Bose Superfluids

The Nambu-Goldstone (NG) modes in a nonrelativistic system can be classified into two types from their characteristic features: being of either an odd (type I) or an even (type II) power energy-momentum dispersion. Conventionally, the type-II NG modes may universally arise from spontaneous breaking of noncommutative symmetry pairs. Here, we predict a novel type of quadratically dispersed NG modes that emerges in mixed s and p band Bose superfluids in an optical lattice and, unlike the conventional type-II NG modes, cannot be solely interpreted with the celebrated symmetry-based argument. Instead, we show that the existence of such modes has a profound connection to the topological transition on projective complex order-parameter space. The detection scheme is also proposed. Our Letter reveals a new universal mechanism for emergence of type-II NG modes, which bridges intrinsically the Landau symmetry-breaking and topological theories.

Ideal Weyl semimetal with 3D spin-orbit coupled ultracold quantum gas

There is an immense effort in search for various types of Weyl semimetals, of which the most fundamental phase consists of the minimal number of i.e. two Weyl points, but is hard to engineer in solids. Here we demonstrate how such fundamental Weyl semimetal can be realized in a maneuverable optical Raman lattice, with which the three-dimensional (3D) spin-orbit (SO) coupling is synthesised for ultracold atoms. In addition, a new novel Weyl phase with coexisting Weyl nodal points and nodal ring is also predicted here, and is shown to be protected by nontrivial linking numbers. We further propose feasible techniques to precisely resolve 3D Weyl band topology through 2D equilibrium and dynamical measurements. This work leads to the first realization of the most fundamental Weyl semimetal band and the 3D SO coupling for ultracold quantum gases, which are respectively the significant issues in the condensed matter and ultracold atom physics.

Novel Quantum Phases of Two-Component Bosons with Pair Hopping in Synthetic Dimension

We study two-component (or pseudospin-1/2) bosons with pair hopping interactions in synthetic dimension, for which a feasible experimental scheme on a square optical lattice is also presented. Previous studies have shown that two-component bosons with on-site interspecies interaction can only generate nontrivial interspecies paired superfluid (super-counter-fluidity or pair-superfluid) states. In contrast, apart from interspecies paired superfluid, we reveal two new phases by considering this additional pair hopping interaction. These novel phases are intraspecies paired superfluid (molecular superfluid) and an exotic noninteger Mott insulator which shows a noninteger atom number at each site for each species, but an integer for total atom number.

Detecting Fractional Chern Insulators in Optical Lattices through Quantized Displacement

The realization of interacting topological states of matter such as fractional Chern insulators (FCIs) in cold atom systems has recently come within experimental reach due to the engineering of optical lattices with synthetic gauge fields providing the required topological band structures. However, detecting their occurrence might prove difficult since transport measurements akin to those in solid state systems are challenging to perform in cold atom setups and alternatives have to be found. We show that for a ν=1/2 FCI state realized in the lowest band of a Harper-Hofstadter model of interacting bosons confined by a harmonic trapping potential, the fractionally quantized Hall conductivity σ_{xy} can be accurately determined by the displacement of the atomic cloud under the action of a constant force which provides a suitable experimentally measurable signal for detecting the topological nature of the state. Using matrix-product state algorithms, we show that, in both cylinder and square geometries, the movement of the particle cloud in time under the application of a constant force field on top of the confining potential is proportional to σ_{xy} for an extended range of field strengths.

Kibble-Zurek Behavior in Disordered Chern Insulators

Even though no local order parameter in the sense of the Landau theory exists for topological quantum phase transitions in Chern insulators, the highly nonlocal Berry curvature exhibits critical behavior near a quantum critical point. We investigate the critical properties of its real space analog, the local Chern marker, in weakly disordered Chern insulators. Because of disorder, inhomogeneities appear in the spatial distribution of the local Chern marker. Their size exhibits power-law scaling with the critical exponent matching the one extracted from the Berry curvature of a clean system. We drive the system slowly through such a quantum phase transition. The characteristic size of inhomogeneities in the nonequilibrium postquench state obeys the Kibble-Zurek scaling. In this setting, the local Chern marker thus does behave in a similar way as a local order parameter for a symmetry breaking second order phase transition. The Kibble-Zurek scaling also holds for the inhomogeneities in the spatial distribution of excitations and of the orbital polarization.

Unified Theory to Characterize Floquet Topological Phases by Quench Dynamics

The conventional characterization of periodically driven systems usually necessitates the time-domain information beyond Floquet bands, hence lacking universal and direct schemes of measuring Floquet topological invariants. Here we propose a unified theory, based on quantum quenches, to characterize generic d-dimensional Floquet topological phases in which the topological invariants are constructed with only minimal information of the static Floquet bands. For a d-dimensional phase that is initially static and trivial, we introduce the quench dynamics by suddenly turning on the periodic driving. We show that the quench dynamics exhibits emergent topological patterns in (d-1)-dimensional momentum subspaces where Floquet bands cross, from which the Floquet topological invariants are directly obtained. This result provides a simple and unified characterization in which one can extract the number of conventional and anomalous Floquet boundary modes and identify the topologically protected singularities in the phase bands. These applications are illustrated with one- and two-dimensional models that are readily accessible in cold-atom experiments. Our study opens a new framework for the characterization of Floquet topological phases.

Non-Abelian generalizations of the Hofstadter model: spin-orbit-coupled butterfly pairs

The Hofstadter model, well known for its fractal butterfly spectrum, describes two-dimensional electrons under a perpendicular magnetic field, which gives rise to the integer quantum Hall effect. Inspired by the real-space building blocks of non-Abelian gauge fields from a recent experiment, we introduce and theoretically study two non-Abelian generalizations of the Hofstadter model. Each model describes two pairs of Hofstadter butterflies that are spin-orbit coupled. In contrast to the original Hofstadter model that can be equivalently studied in the Landau and symmetric gauges, the corresponding non-Abelian generalizations exhibit distinct spectra due to the non-commutativity of the gauge fields. We derive the genuine (necessary and sufficient) non-Abelian condition for the two models from the commutativity of their arbitrary loop operators. At zero energy, the models are gapless and host Weyl and Dirac points protected by internal and crystalline symmetries. Double (8-fold), triple (12-fold), and quadrupole (16-fold) Dirac points also emerge, especially under equal hopping phases of the non-Abelian potentials. At other fillings, the gapped phases of the models give rise to topological insulators. We conclude by discussing possible schemes for experimental realization of the models on photonic platforms.

Quantum Phases of Three-Dimensional Chiral Topological Insulators on a Spin Quantum Simulator

The detection of topological phases of matter has become a central issue in recent years. Conventionally, the realization of a specific topological phase in condensed matter physics relies on probing the underlying surface band dispersion or quantum transport signature of a real material, which may be imperfect or even absent. On the other hand, quantum simulation offers an alternative approach to directly measure the topological invariant on a universal quantum computer. However, experimentally demonstrating high-dimensional topological phases remains a challenge due to the technical limitations of current experimental platforms. Here, we investigate the three-dimensional topological insulators in the AIII (chiral unitary) symmetry class, which yet lack experimental realization. Using the nuclear magnetic resonance system, we experimentally demonstrate their topological properties, where a dynamical quenching approach is adopted and the dynamical bulk-boundary correspondence in the momentum space is observed. As a result, the topological invariants are measured with high precision on the band-inversion surface, exhibiting robustness to the decoherence effect. Our Letter paves the way toward the quantum simulation of topological phases of matter in higher dimensions and more complex systems through controllable quantum phases transitions.

Realization and Detection of Nonergodic Critical Phases in an Optical Raman Lattice

The critical phases, being delocalized but nonergodic, are fundamental phases different from both the many-body localization and ergodic extended quantum phases, and have so far not been realized in experiment. Here we propose an incommensurate topological insulating model of AIII symmetry class to realize such critical phases through an optical Raman lattice scheme, which possesses a one-dimensional (1D) spin-orbit coupling and an incommensurate Zeeman potential. We show the existence of both noninteracting and many-body critical phases, which can coexist with the topological phase, and show that the critical-localization transition coincides with the topological phase boundary in noninteracting regime. The dynamical detection of the critical phases is proposed and studied in detail based on the available experimental techniques. Finally, we demonstrate how the proposed critical phases can be achieved within the current ultracold atom experiments. This work paves the way to observe the novel critical phases.